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Language of Mathematics: Relative or Absolute?


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#1 Comandante

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Posted 28 November 2008 - 09:40 AM

This is a question about the nature of Mathematics as a language. The question is very simple, as the thread name suggests, but I’ll expand my own perception of it here and so it would be helpful if someone can leave a few comments.

Is mathematics relative or absolute? My insight developed as follows…; If Mathematics as language was invented by us it was definitely invented based on the observations of the physical space and laws on our planet. To simplify; we can find one thing in common between 2 yellow stones on one location and then 2 blue stones on another – the fact that there are 2 stones in each group. So number 2 is the commonality. If we add these 2 groups of stones together then we get 4 stones in a group. Same goes with almost any other similar situation.

However, one exception would be that if we have 2 protons in one location and 2 antiprotons in another, surely we can still count 4 protons but they can never coexist together in contact with each other. So why is it that 2+2 is still equal to 4, wouldn’t that statement be meaningless if we tried to count the particles by using mathematics without actually knowing that they annihilate?

Now imagine that yellow and blue stones could never have coexisted (for the purpose of the argument) in the same group in contact with each other, how would we be able to say that 2 yellow stones and 2 blue stones would be 4 stones? Wouldn’t we as humans have developed a different rule for the situation and say that 2+2 = 0? I came to conclude that the problem can be fixed by marking one group of particles with an opposite charge (which is what is generally done), but doesn’t this then become merely an adjustment (or contamination) in mathematics? Why did we have to add physical characteristics as extra notations to everything that didn’t make sense otherwise? These issues have me wondering whether mathematics as we know it can be entirely different in other parts of the universe(s)… so if someone can shed some light it would be great.
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#2 Gilded

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Posted 28 November 2008 - 10:07 AM

About the proton thing, there will be 4 gamma photons to count, just like there would be a certain mass to measure even if you grind the 4 stones into rubble. There's always something to count or measure, no matter where in the universe you might live. :)

Here's the Wikipedia article about how mathematics came to be:
http://en.wikipedia...._of_mathematics

While mathematics has stemmed from observations of reality like everything else really, I'd say it's extraordinarily context free, seeing how certain cultures independently came up with concepts like infinity and zero. Obviously, there are different base numbers and numeric systems in general, but mathematics is a study and measurement of reality in perhaps its purest form, and is certainly a concept any successful and developed species is likely to discover.

Then again, on the topic of context, there is theoretical mathematics which is just batsh!t insane and has nothing to do with anything else. :D
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#3 Comandante

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Posted 28 November 2008 - 11:59 AM

Thanks. Hmm.. Well, that's certainly law of conservation of energy, surely we can't destroy energy and it'll be the same on both sides of equation, but I'm still confused with what I mentioned in post #1.

Certain cultures came up with same ideas because they were exposed to the same reality. What I'm saying is, if the things didn't work the same some place else then it follows that their maths would be different and adjusted to work with their physical laws accordingly, wouldn't it?

edit: misread that part with zero and infinity. That's certainly a good point, I'll think about it for a while.
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#4 the tree

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Posted 28 November 2008 - 06:49 PM

What I'm saying is, if the things didn't work the same some place else then it follows that their maths would be different and adjusted to work with their physical laws accordingly, wouldn't it?

Maybe really big beings would have involved the effects of relativity in their first steps into mechanics, and come out with a pretty different calculus., possibly. But you've got to consider that a lot of our maths has nothing to do with the real world at all.
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#5 ajb

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Posted 28 November 2008 - 06:58 PM

. But you've got to consider that a lot of our maths has nothing to do with the real world at all.


Interesting thought. Can you name some areas that have not be of use in physics or applied mathematics?
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#6 npts2020

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Posted 30 November 2008 - 02:07 PM

Wouldn't 2 protons plus 2 antiprotons add up to zero because you are really adding 2+(-)2=0. Unlike the color of a rock, charge is an intrinsic property of a proton or antiproton and cannot be changed without changing the particle.
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#7 the tree

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Posted 30 November 2008 - 08:56 PM

Interesting thought. Can you name some areas that have not be of use in physics or applied mathematics?

AFAIK, Category Theory has no practical application. Although a lot of the time, something turns out to be of use well after it's developed, so it goes a long time having no real application but not indefinitely, such as complex analysis.
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#8 Comandante

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Posted 2 December 2008 - 05:49 AM

Wouldn't 2 protons plus 2 antiprotons add up to zero because you are really adding 2+(-)2=0. Unlike the color of a rock, charge is an intrinsic property of a proton or antiproton and cannot be changed without changing the particle.


Sure they'll add to 0, however you're misunderstanding my approach to the situation. I was using protons/antiprotons as example, but it could really be any other from infinity of possible situations, we may not know since we aren't accustomed well with laws very very far away. But we can stick to the protons for the time being, until someone comes up with a better example, and if.

So let's say somewhere else in the universe, a civilization somehow has the ability to detect these subatomic particles and becomes greatly interested in them, so we're talking about primitive time, perhaps to correlate to situation on Earth about 2000BC. And as they become smarter they realize that two particles coming in contact (proton and antiproton) would give nothing. Say they don't know that energy is released and they can only observe 2 particles (apparently the same) colliding and annihilating. Wouldn't their means of noting this be something like 1 + 1 = 0 ? Now, how long do you think it would take them until they started adding up apples and stones using the same maths? They would definitely need to adjust it once they did, for it would be extremely unnatural for them, after being subjected to the idea that 1+1 = 0 for so many years, to actually change that thought and say that 1+1 = 2. Now, wouldn't they rather add extra notation to the fact that 1+1 = 2 than to their original idea (1+1= 0). I imagine they would add small characters or symbols to both number 1's so that it makes sense, something like 1a+1s=2as (a = apple, s = stone).

That said, let's just take a look at that line again; 1a+1s = 2as. If they were to send us a letter in a spaceship to tell us about their world, what would we make up of those numbers? Wouldn't we claim that their maths is seemingly different to ours? And is it? That's what my question was. (That's assuming they were using similar numbering system!)

Also, please bear in mind that we're now looking into the idea of mathematics as a tool for working with numbers, rather than the idea of physical characteristics and capabilities of various objects throughout the universe. What I really want is for people to disprove everything I said so that I can rest peacefully in my chair when I start studying maths again :D
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#9 ajb

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Posted 2 December 2008 - 12:46 PM

AFAIK, Category Theory has no practical application. Although a lot of the time, something turns out to be of use well after it's developed, so it goes a long time having no real application but not indefinitely, such as complex analysis.



Category theory is becoming more and more used in string/M-theory to describe branes. Also, it is a very economical language to describe things and as such does appear in mathematical and theoretical physics papers.

There is a drive for the categorification of physics. Which mostly means work outside the category of sets. This is important in modern physics.

Not that I am a huge fan of category theory, but a little can go a long way.
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#10 D H

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Posted 2 December 2008 - 12:55 PM

Sysco,

You are trying to ascribe physical reality to mathematics. That is a bad idea, even with something as basic as counting. One drop of water added to one drop of water makes one (bigger) drop of water, one proton colliding with one antiproton results in zero protons, and one bottle colliding with one bottle makes a lot of shards of glass. None of these invalidate 1+1=2. Finding that space itself is quantized will not invalidate the mathematical concept of a point. Finding that the universe is non-Euclidean (if that happens) will not invalidate mathematical concept of Euclidean geometry. Euclid's geometry will still be as true then as it is now and as it was in Euclid's time.

That final point about Euclidean geometry illustrates a key difference between mathematics and science. Mathematical theorems, once proved true, remain true forever. Scientific theories are at best provisionally true. You are conflating mathematics with the use of mathematics as a tool by scientists, economists, etc. Don't do that!

=======================

AFAIK, Category Theory has no practical application.


Daniel S. Freed, "Higher Algebraic Structures and Quantization", preprint at http://xxx.lanl.gov/abs/hep-th/9212115

Louis Crane, "Clock and Category; IS QUANTUM GRAVITY ALGEBRAIC", preprint at http://xxx.lanl.gov/abs/gr-qc/9504038

John Baez wrote an extended "Tale of n-Categories" beginning in http://math.ucr.edu/...aez/week73.html and ending in http://math.ucr.edu/...ez/week100.html
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#11 Comandante

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Posted 2 December 2008 - 02:36 PM

Sysco,

You are conflating mathematics with the use of mathematics as a tool by scientists, economists, etc. Don't do that!


That's probably right. I'll try to think of a better way of putting it, no promises on that one though.

So, is your definitive opinion that whole of mathematics is true for the entire universe (and if more, in other universes)?
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#12 D H

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Posted 2 December 2008 - 02:48 PM

So, is your definitive opinion that whole of mathematics is true for the entire universe (and if more, in other universes)?

No! First off, that is conflating mathematics with physical reality. Secondly, the whole of mathematics cannot be proven true. Gödel's incompleteness theorems get in the way.
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#13 Comandante

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Posted 2 December 2008 - 02:54 PM

the whole of mathematics cannot be proven true. Gödel's incompleteness theorems get in the way.


Never heard of that guy. Maybe I'll come across him next year.

So, if I'm getting this right, of the mathematics that CAN be proven, such as 1+1, it is definitely the same and correct everywhere, i.e., it is absolute?
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#14 the tree

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Posted 2 December 2008 - 06:21 PM

So, if I'm getting this right, of the mathematics that CAN be proven, such as 1+1, it is definitely the same and correct everywhere, i.e., it is absolute?

No. It's only correct because of the axioms constructed for arithmetic which 1) there are many versions of 2) cannot be proven to be internally consistent. Point (2) is Gödel's incompleteness theorem.
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#15 ajb

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Posted 14 December 2008 - 04:37 PM

This whole discussion make me "conjecture" that all abstract mathematics is physically realised.

What I mean is that all abstract mathematics can be found in nature via "physical theories". I know this is a very lose statement, but it goes back to my point about finding a branch of mathematics that has not be of some use in theoretical physics and applied mathematics. It makes me ponder just how abstract is mathematics? Or put another way, is there really such a thing as pure mathematics?

With category theory in mind I have ordered a copy of Geroch, Mathematical Physics, Chicago University Press (1 Nov 1985).

Edited by ajb, 14 December 2008 - 04:47 PM.

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#16 the tree

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Posted 14 December 2008 - 06:09 PM

There is an obvious gap between the time when the mathematics is thought up, and the physical realisation. Example being complex arithmetic which was developed and had become established way before electrical engineering or fluid dynamics. However, if you tack an 'eventually' onto the end of that conjecture it looses all it's meaning.
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#17 ajb

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Posted 14 December 2008 - 08:17 PM

Of course what I have said is purposely vague, but I don't think I want to "attach eventually". By this you mean if we wait long enough someone will find an application?

But isn't that a "conjecture" itself which is equivalent to mine?

This is by no means a strict metatheorem or anything like that, just a loose though.
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#18 D H

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Posted 14 December 2008 - 08:52 PM

This whole discussion make me "conjecture" that all abstract mathematics is physically realised.

Suppose we discover that not only matter is discrete, but so is space-time. Said discovery will make calculus and the reals physically non-realizable. We will still be able to use calculus and the reals to describe the universe, but only in an approximate sense.

Full disclosure: I am of the opinion that mathematics is invented, not discovered. However, because we humans have rather limited imaginations, most of our mathematical inventions are motivated by physical reality.

That said, Cantor had a dang near unbounded imagination. What, pray tell, would the physical realization of infinitary logic look like?
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#19 ajb

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Posted 14 December 2008 - 09:11 PM

Indeed in comes down to the notion of "physically realisable" or some similar idea. However, I am sure you understand what I am saying, even if it is an extremely lose idea. Lets define a related notion. Lets say it is physics if there has been at least one paper published by someone in a physics department.

I find it very interesting that to my knowledge just about all known mathematics is "useful" somewhere. Your point on physics being a huge influence on mathematics is well noted.

Logic and topos theory do enter in theoretical physics, particularly in attempts to quantise gravity and of course in computer science. As for infinitary logic I don't know. I also wonder about other areas of proof theory and model theory. I am confident someone has applied them somewhere. This could be a fun exercise tracking down references.

Edited by ajb, 14 December 2008 - 09:38 PM.

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